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In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.
In the remainder of the article it is assumed that the graph is represented using an adjacency matrix. We expect the output of the algorithm to be a distancematrix D {\displaystyle D} . In D {\displaystyle D} , every entry d i , j {\displaystyle d_{i,j}} is the weight of the shortest path in G {\displaystyle G} from node i {\displaystyle i} to ...
As Itai & Rodeh (1978) observe, the graph contains a triangle if and only if its adjacency matrix and the square of the adjacency matrix contain nonzero entries in the same cell. Therefore, fast matrix multiplication techniques can be applied to find triangles in time O (n 2.376).
The Floyd–Warshall algorithm can be used to solve the following problems, among others: Shortest paths in directed graphs (Floyd's algorithm). Transitive closure of directed graphs (Warshall's algorithm). In Warshall's original formulation of the algorithm, the graph is unweighted and represented by a Boolean adjacency matrix.
In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices.
In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A. [1] Some other similar mathematical objects are also called "adjacency algebra".
Here W is an adjacency matrix which represents the states of fixed policy MDP which forms an undirected graph (N,E). D is a diagonal matrix related to nodes' degrees. In discrete state space, the adjacency matrix W {\displaystyle W} could be constructed by simply checking whether two states are connected, and D could be calculated by summing up ...
The incidence matrix of an incidence structure C is a p × q matrix B (or its transpose), where p and q are the number of points and lines respectively, such that B i,j = 1 if the point p i and line L j are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure.